# scipy.linalg.ordqz#

scipy.linalg.ordqz(A, B, sort='lhp', output='real', overwrite_a=False, overwrite_b=False, check_finite=True)[source]#

QZ decomposition for a pair of matrices with reordering.

Parameters:
A(N, N) array_like

2-D array to decompose

B(N, N) array_like

2-D array to decompose

sort{callable, ‘lhp’, ‘rhp’, ‘iuc’, ‘ouc’}, optional

Specifies whether the upper eigenvalues should be sorted. A callable may be passed that, given an ordered pair ```(alpha, beta)``` representing the eigenvalue `x = (alpha/beta)`, returns a boolean denoting whether the eigenvalue should be sorted to the top-left (True). For the real matrix pairs `beta` is real while `alpha` can be complex, and for complex matrix pairs both `alpha` and `beta` can be complex. The callable must be able to accept a NumPy array. Alternatively, string parameters may be used:

• ‘lhp’ Left-hand plane (x.real < 0.0)

• ‘rhp’ Right-hand plane (x.real > 0.0)

• ‘iuc’ Inside the unit circle (x*x.conjugate() < 1.0)

• ‘ouc’ Outside the unit circle (x*x.conjugate() > 1.0)

With the predefined sorting functions, an infinite eigenvalue (i.e., `alpha != 0` and `beta = 0`) is considered to lie in neither the left-hand nor the right-hand plane, but it is considered to lie outside the unit circle. For the eigenvalue `(alpha, beta) = (0, 0)`, the predefined sorting functions all return False.

outputstr {‘real’,’complex’}, optional

Construct the real or complex QZ decomposition for real matrices. Default is ‘real’.

overwrite_abool, optional

If True, the contents of A are overwritten.

overwrite_bbool, optional

If True, the contents of B are overwritten.

check_finitebool, optional

If true checks the elements of A and B are finite numbers. If false does no checking and passes matrix through to underlying algorithm.

Returns:
AA(N, N) ndarray

Generalized Schur form of A.

BB(N, N) ndarray

Generalized Schur form of B.

alpha(N,) ndarray

alpha = alphar + alphai * 1j. See notes.

beta(N,) ndarray

See notes.

Q(N, N) ndarray

The left Schur vectors.

Z(N, N) ndarray

The right Schur vectors.

`qz`

Notes

On exit, `(ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N`, will be the generalized eigenvalues. `ALPHAR(j) + ALPHAI(j)*i` and `BETA(j),j=1,...,N` are the diagonals of the complex Schur form (S,T) that would result if the 2-by-2 diagonal blocks of the real generalized Schur form of (A,B) were further reduced to triangular form using complex unitary transformations. If ALPHAI(j) is zero, then the jth eigenvalue is real; if positive, then the ```j``th and ``(j+1)``st eigenvalues are a complex conjugate pair, with ``ALPHAI(j+1)``` negative.

New in version 0.17.0.

Examples

```>>> import numpy as np
>>> from scipy.linalg import ordqz
>>> A = np.array([[2, 5, 8, 7], [5, 2, 2, 8], [7, 5, 6, 6], [5, 4, 4, 8]])
>>> B = np.array([[0, 6, 0, 0], [5, 0, 2, 1], [5, 2, 6, 6], [4, 7, 7, 7]])
>>> AA, BB, alpha, beta, Q, Z = ordqz(A, B, sort='lhp')
```

Since we have sorted for left half plane eigenvalues, negatives come first

```>>> (alpha/beta).real < 0
array([ True,  True, False, False], dtype=bool)
```